Chapter 6.6, Problem 9E

To determine

To calculate: .The equation for the value of $x$ .

Answer to Problem 9E

The required value of $x$ is $1.771$ .

Explanation of Solution

Given Information:

The given equation is $3^x=7$ .

Formula Used Power rule of logarithmic functions:

  ${\log }_a{x}^p=p{\log }_{a}x$

Where $a$ is the base, $x$ is the variable and $p$ is the power of variable.

Calculation:

Consider the equation $3^x=7$ .

Take common logarithm with base $10$ of the above equation $3^x=7$ .

  ${\log }_{10}{3}^x={\log }_{10}7$

Now, use power rule of logarithm ${\log }_a{x}^p=p{\log }_{a}x$ in the above equation.

  $x{\log }_{10}3={\log }_{10}7$

Divide each side by ${\log }_{10}3$ of the above equation.

  $\begin{array}{c}\frac{x{\log }_{10}3}{{\log }_{10}3}=\frac{{\log }_{10}7}{{\log }_{10}3}\\ x=\frac{{\log }_{10}7}{{\log }_{10}3}\end{array}$

Substitute $0.845$ for ${\log }_{10}7$ and $0.477$ for ${\log }_{10}3$ in the above equations.

  $\begin{array}{c}x=\frac{0.845}{0.477}\\ x=1.771\end{array}$

Hence, the required value of $x$ is $1.771$ .